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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | import OAI.Analysis.FiniteFactor.OrbitSampling.RectangleError
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
/-- Actual rational rectangle masses approach their ideal product integrals
on a single conull event. This is the stochastic approximation step; the
identification of the ideal limiting law is a separate obligation. -/
theorem sourceSampling_rectangle_approximation_event :
∀ᵐ θ ∂sourceAngleLaw, ∀ (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest),
Tendsto (fun stage =>
((sourceSamplingUntrimmed I stage θ)
(sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T))).toReal -
∫ x, sourceSamplingIdealDensity stage false (fun i => decide (i ∈ I))
(sourceSamplingRationalX T) (sourceSamplingRationalY T) x ∂sourceSamplingHaar)
atTop (𝓝 0) := by
filter_upwards [sourceSampling_rational_event] with θ hθ
intro I T
have he := (hθ I false T).1
have hs := Real.continuous_sqrt.continuousAt.tendsto.comp he
simp only [Real.sqrt_zero] at hs
apply tendsto_zero_iff_norm_tendsto_zero.mpr
simp only [Real.norm_eq_abs]
apply squeeze_zero' (Eventually.of_forall (fun stage => abs_nonneg _)) _ hs
filter_upwards [sourceSamplingPrefix_eventually_covers T.1] with stage hstage
exact Real.abs_le_sqrt (sourceSamplingUntrimmed_rectangle_error I stage
(sourceSamplingRationalX T) (sourceSamplingRationalY T)
(fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc)
(sourceSamplingRationalTests_outside T stage hstage) θ)
end
end FiniteFactor
end OAI
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